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On the Bonsall cone spectral radius and the approximate point spectrum
Strichartz estimates for $N$-body Schrödinger operators with small potential interactions
Center for Mathematical Analysis & Computation (CMAC), Yonsei University, Seoul 03722, Korea |
In this paper, we prove Strichartz estimates for $N$-body Schrödinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the $V_S^p$-norm of Koch and Tataru [
References:
[1] |
M. Beceanu and M. Goldberg,
Schrödinger dispersive estimates for a scaling-critical class of
potentials, Comm. Math. Phys., 314 (2012), 471-481.
doi: 10.1007/s00220-012-1435-x. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp. |
[3] |
N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh,
Strichartz estimates for the wave
and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.
doi: 10.1016/S0022-1236(03)00238-6. |
[4] |
N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh,
Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680.
doi: 10.1512/iumj.2004.53.2541. |
[5] |
T. Chen, Y. Hong and N. Pavlović,
Global well-posedness of the NLS system for infinitely
many fermions, Arch. Ration. Mech. Anal., 224 (2017), 91-123.
doi: 10.1007/s00205-016-1068-x. |
[6] |
T. Chen and N. Pavlović,
Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from
manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588.
doi: 10.1007/s00023-013-0248-6. |
[7] |
X. Chen and J. Holmer,
On the Klainerman-Machedon conjecture for the quantum BBGKY
hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016), 1161-1200.
doi: 10.4171/JEMS/610. |
[8] |
D. Fujiwara,
Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600.
doi: 10.1215/S0012-7094-80-04734-1. |
[9] |
M. Goldberg,
Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006), 731-750.
doi: 10.1353/ajm.2006.0025. |
[10] |
M. Goldberg,
Dispersive bounds for the three-dimensional Schrödinger equation with almost
critical potentials, Geom. Funct. Anal., 16 (2006), 517-536.
doi: 10.1007/s00039-006-0568-5. |
[11] |
M. Goldberg and W. Schlag,
Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178.
doi: 10.1007/s00220-004-1140-5. |
[12] |
G. Graf,
Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990), 73-101.
doi: 10.1007/BF02278000. |
[13] |
M. Hadac, S. Herr and H. Koch,
Well-posedness and scattering for the KP-Ⅱ equation in a
critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[14] |
S. Herr, D. Tataru and N. Tzvetkov,
Global well-posedness of the energy-critical nonlinear
Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011), 329-349.
doi: 10.1215/00127094-1415889. |
[15] |
J.-L. Journe, A. Soffer and C. Sogge,
Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.
doi: 10.1002/cpa.3160440504. |
[16] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[17] |
K. Kirkpatrick, B. Schlein and G. Staffilani,
Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130.
doi: 10.1353/ajm.2011.0004. |
[18] |
S. Klainerman and M. Machedon,
On the uniqueness of solutions to the Gross-Pitaevskii
hierarchy, Comm. Math. Phys., 279 (2008), 169-185.
doi: 10.1007/s00220-008-0426-4. |
[19] |
H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp.
doi: 10.1093/imrn/rnm053. |
[20] |
H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014).
doi: 10.1007/978-3-0348-0736-4. |
[21] |
I. Rodnianski and W. Schlag,
Time decay for solutions of Schrödinger equations with rough
and time-dependent potentials, Invent. Math., 155 (2004), 451-513.
doi: 10.1007/s00222-003-0325-4. |
[22] |
W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects
of nonlinear dispersive equations, Ann. of Math. Stud., Princeton Univ. Press, 163 (2007),
255–285. |
[23] |
I. M. Sigal and A. Soffer,
The N-particle scattering problem: Asymptotic completeness for
short-range systems, Ann. of Math.(3), 126 (1987), 35-108.
doi: 10.2307/1971345. |
show all references
References:
[1] |
M. Beceanu and M. Goldberg,
Schrödinger dispersive estimates for a scaling-critical class of
potentials, Comm. Math. Phys., 314 (2012), 471-481.
doi: 10.1007/s00220-012-1435-x. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp. |
[3] |
N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh,
Strichartz estimates for the wave
and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.
doi: 10.1016/S0022-1236(03)00238-6. |
[4] |
N. Burq, F. Planchon, J. Stalker and S. Tahvildar-Zadeh,
Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680.
doi: 10.1512/iumj.2004.53.2541. |
[5] |
T. Chen, Y. Hong and N. Pavlović,
Global well-posedness of the NLS system for infinitely
many fermions, Arch. Ration. Mech. Anal., 224 (2017), 91-123.
doi: 10.1007/s00205-016-1068-x. |
[6] |
T. Chen and N. Pavlović,
Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from
manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588.
doi: 10.1007/s00023-013-0248-6. |
[7] |
X. Chen and J. Holmer,
On the Klainerman-Machedon conjecture for the quantum BBGKY
hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016), 1161-1200.
doi: 10.4171/JEMS/610. |
[8] |
D. Fujiwara,
Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600.
doi: 10.1215/S0012-7094-80-04734-1. |
[9] |
M. Goldberg,
Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006), 731-750.
doi: 10.1353/ajm.2006.0025. |
[10] |
M. Goldberg,
Dispersive bounds for the three-dimensional Schrödinger equation with almost
critical potentials, Geom. Funct. Anal., 16 (2006), 517-536.
doi: 10.1007/s00039-006-0568-5. |
[11] |
M. Goldberg and W. Schlag,
Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178.
doi: 10.1007/s00220-004-1140-5. |
[12] |
G. Graf,
Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990), 73-101.
doi: 10.1007/BF02278000. |
[13] |
M. Hadac, S. Herr and H. Koch,
Well-posedness and scattering for the KP-Ⅱ equation in a
critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[14] |
S. Herr, D. Tataru and N. Tzvetkov,
Global well-posedness of the energy-critical nonlinear
Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011), 329-349.
doi: 10.1215/00127094-1415889. |
[15] |
J.-L. Journe, A. Soffer and C. Sogge,
Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.
doi: 10.1002/cpa.3160440504. |
[16] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[17] |
K. Kirkpatrick, B. Schlein and G. Staffilani,
Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130.
doi: 10.1353/ajm.2011.0004. |
[18] |
S. Klainerman and M. Machedon,
On the uniqueness of solutions to the Gross-Pitaevskii
hierarchy, Comm. Math. Phys., 279 (2008), 169-185.
doi: 10.1007/s00220-008-0426-4. |
[19] |
H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp.
doi: 10.1093/imrn/rnm053. |
[20] |
H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014).
doi: 10.1007/978-3-0348-0736-4. |
[21] |
I. Rodnianski and W. Schlag,
Time decay for solutions of Schrödinger equations with rough
and time-dependent potentials, Invent. Math., 155 (2004), 451-513.
doi: 10.1007/s00222-003-0325-4. |
[22] |
W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects
of nonlinear dispersive equations, Ann. of Math. Stud., Princeton Univ. Press, 163 (2007),
255–285. |
[23] |
I. M. Sigal and A. Soffer,
The N-particle scattering problem: Asymptotic completeness for
short-range systems, Ann. of Math.(3), 126 (1987), 35-108.
doi: 10.2307/1971345. |
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